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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 5040.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5040.bc1 | 5040bl3 | \([0, 0, 0, -2712, -50209]\) | \(189123395584/16078125\) | \(187535250000\) | \([2]\) | \(6912\) | \(0.90451\) | |
| 5040.bc2 | 5040bl1 | \([0, 0, 0, -552, 4979]\) | \(1594753024/4725\) | \(55112400\) | \([2]\) | \(2304\) | \(0.35520\) | \(\Gamma_0(N)\)-optimal |
| 5040.bc3 | 5040bl2 | \([0, 0, 0, -327, 9074]\) | \(-20720464/178605\) | \(-33331979520\) | \([2]\) | \(4608\) | \(0.70178\) | |
| 5040.bc4 | 5040bl4 | \([0, 0, 0, 2913, -231334]\) | \(14647977776/132355125\) | \(-24700642848000\) | \([2]\) | \(13824\) | \(1.2511\) |
Rank
sage: E.rank()
The elliptic curves in class 5040.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 5040.bc do not have complex multiplication.Modular form 5040.2.a.bc
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
